Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 and 4cm (2 sides of rect. along legs).

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lemjay | High School Teacher | (Level 3) Senior Educator

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To solve, refer to the figure below.

Let the length and width of the inscribed rectangle be l and w. So, its area is:

`A = l * w`

To express this as one variable, let's apply similar triangles. Set the ratio of height to base of the big triangle equal to the ration of height to base of the small triangle.

3/4=l/(4-w)

Then, isolate the l.

`3/4(4-w)=l`

`3-3/4w=l`

Plug-in this to the area of the rectangle.

`A=l*w = (3-3/4w)w=3w -3/4w^2`

`A=-3/4w^2+3w`

Notice that the expression that represents the area of the rectangle is in quadratic form. Take note that the graph of a quadratic function is a parabola. 

If we graph our area,  it will be:

Notice that its vertex is the maximum point. Referring to the graph, its vertex is (2,3). The y-coordinate of this point refers to the area of the rectangle.

Therefore, the area of the largest rectangle that can be inscribed in the given right triangle is 3 square centimeter.

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